The Colour Group

We can postulate the existence of an exact SU(Nc) colour symmetry with Nc = 3 colours. Let U be a matrix belonging to the group SU(Nc). Then:

$\text{Special: }det(U)=1$

$\text{Unitary: }U{U}^{†}=U^{†}U=1$

We want to construct mesons and baryons as colour singlets, i.e., they should have no isolated colour charges and transform trivially under the colour group. How?

• Mesons: ${\bar{q}}^i{q}^i\to {\bar{q}}^j(U^{†}{)}_{ji}{U}_{ik}{q}^k={\bar{q}}^i{q}^i$
• Baryons: ${ϵ}_{ijk}{q}^i{q}^j{q}^k=\underset{det(U){ϵ}_{i^{\prime }{j}^{\prime }{k}^{\prime }}}{\underbrace{{ϵ}_{ijk}{U}_{i{i}^{\prime }}{U}_{j{j}^{\prime }}{U}_{k{k}^{\prime }}}}{q}^{i^{\prime }}{q}^{j^{\prime }}{q}^{k^{\prime }}={ϵ}_{ijk}{q}^i{q}^j{q}^k$

Many events that take place at the Large Hadron Collider (LHC), a high-energy particle collider, are governed by QCD.